Upload
others
View
3
Download
0
Embed Size (px)
Citation preview
PacificJournal ofMathematics
STRONG KÄHLER WITH TORSION STRUCTURESFROM ALMOST CONTACT MANIFOLDS
MARISA FERNÁNDEZ, ANNA FINO,LUIS UGARTE AND RAQUEL VILLACAMPA
Volume 249 No. 1 January 2011
PACIFIC JOURNAL OF MATHEMATICSVol. 249, No. 1, 2011
STRONG KÄHLER WITH TORSION STRUCTURESFROM ALMOST CONTACT MANIFOLDS
MARISA FERNÁNDEZ, ANNA FINO,LUIS UGARTE AND RAQUEL VILLACAMPA
For an almost contact metric manifold N , we find conditions under whicheither the total space of an S1-bundle over N or the Riemannian cone overN admit a strong Kähler with torsion (SKT) structure. In so doing, weconstruct new 6-dimensional SKT manifolds. Moreover, we study the geo-metric structure induced on a hypersurface of an SKT manifold and use itto construct new SKT manifolds via appropriate evolution equations. Wealso study hyper-Kähler with torsion (HKT) structures on the total space ofan S1-bundle over manifolds with three almost contact structures.
1. Introduction
On any Hermitian manifold (M2n, J, h) there exists a unique Hermitian connection∇
B with totally skew-symmetric torsion, which is called the Bismut connectionafter [Bismut 1989]. The torsion 3-form h(X, T B(Y, Z)) of ∇B can be identifiedwith the 3-form
−J dF( · , · , · )=−dF(J · , J · , J · ),
where F( · , · ) = h( · , J · ) is the fundamental 2-form associated to the Hermitianstructure (J, h).
Hermitian structures with closed J dF are called strong Kähler with torsion (inshort, SKT) or pluriclosed [Egidi 2001]. Since ∂∂ acts as 1
2 d J d on forms ofbidegree (1, 1), the latter condition is equivalent to ∂∂F = 0. SKT structures havebeen recently studied by many authors, and they also have applications in type IIstring theory and in 2-dimensional supersymmetric σ -models [Gates et al. 1984;Strominger 1986; Ivanov and Papadopoulos 2001].
The class of SKT metrics includes of course the Kahler metrics, but as in [Finoet al. 2004], we are interested on non-Kahler geometry, so by an SKT metric we
MSC2000: primary 53C55; secondary 53C26, 22E25, 53C15.Keywords: Hermitian metric, hypercomplex structure, torsion, almost contact structure,
Riemannian cone.Partially supported through Project MICINN (Spain) MTM2008-06540-C02-01/02, Project MIUR“Riemannian Metrics and Differentiable Manifolds” and by GNSAGA of INdAM.
49
50 MARISA FERNÁNDEZ, ANNA FINO, LUIS UGARTE AND RAQUEL VILLACAMPA
will mean a Hermitian metric h whose fundamental 2-form F is ∂∂-closed but notd-closed.
Gauduchon [1984] showed that on a compact complex surface, an SKT metriccan be found in the conformal class of any given Hermitian metric, but in higherdimensions the situation is more complicated.
SKT structures on 6-dimensional nilmanifolds, that is, on compact quotients ofnilpotent Lie groups by discrete subgroups, were classified in [Fino et al. 2004;Ugarte 2007]. Simply connected examples of 6-dimensional SKT manifolds havebeen found in [Grantcharov et al. 2008] by using torus bundles, and recently Swann[2010] has reproduced them via the twist construction, by extending them to higherdimensions and finding new compact simply connected SKT manifolds. Moreover,Fino and Tomassini [2009] showed that the SKT condition is preserved by theblow-up construction.
The odd-dimensional analogue of a Hermitian structure is given by a normalalmost contact metric structure. Indeed, on the product N 2n+1
×R of a (2n+1)-dimensional almost contact metric manifold N 2n+1 by the real line R, it is possibleto define a natural almost complex structure, which is integrable if and only ifthe almost contact metric structure on N 2n+1 is normal [Sasaki and Hatakeyama1961]. More generally, it is possible to construct Hermitian manifolds starting froman almost contact metric manifold N 2n+1 by considering a principal fiber bundle Pwith base space N 2n+1 and structural group S1, that is, an S1-bundle over N 2n+1;see [Ogawa 1963]. Indeed, by using the almost contact metric structure on N 2n+1
and the connection 1-form θ , Ogawa constructed an almost Hermitian structure(J, h) on P and found conditions under which J is integrable and (J, h) is Kahler.
In Section 2, we determine in Theorem 2.3 general conditions under which anS1-bundle over an almost contact metric (2n+1)-dimensional manifold N 2n+1 isSKT. We study the particular case when N 2n+1 is quasi-Sasakian, that is, when ithas an almost contact metric structure for which the fundamental form is closed(Corollary 2.4). In this way, we are able to construct some new 6-dimensional SKTexamples, starting from 5-dimensional quasi-Sasakian Lie algebras, and also fromSasakian ones.
A Sasakian structure can be also seen as the analogue, in odd dimensions, ofa Kahler structure. Indeed, by [Boyer and Galicki 1999], a Riemannian manifold(N 2n+1, g) of odd dimension 2n+1 admits a compatible Sasakian structure if andonly if the Riemannian cone N 2n+1
× R+ is Kahler. In Section 3, Theorem 3.1gives the conditions that must be satisfied by the compatible almost contact metricstructure on a Riemannian manifold (N 2n+1, g) in order that the Riemannian coneN 2n+1
×R+ be SKT. We provide an example of an SKT manifold constructed asa Riemannian cone, and in Section 4 we consider the case when the Riemanniancone is 6-dimensional. This case is interesting since one can impose that the SKT
STRONG KÄHLER WITH TORSION STRUCTURES 51
structure is in addition an SKT SU(3)-structure, and one can find relations withthe SU(2)-structures studied by Conti and Salamon [2007].
In Section 5, we study the geometric structure induced naturally on any orientedhypersurface N 2n+1 of a (2n+2)-dimensional manifold M2n+2 carrying an SKTstructure, and in Section 6, we use such structures in Theorem 6.4 to constructnew SKT manifolds via appropriate evolution equations [Hitchin 2001; Conti andSalamon 2007] starting from a 5-dimensional manifold endowed with an SU(2)-structure.
A good quaternionic analogue of Kahler geometry is given by hyper-Kählerwith torsion (in short, HKT) geometry. An HKT manifold is a hyper-Hermitianmanifold (M4n, J1, J2, J3, h) admitting a hyper-Hermitian connection with totallyskew-symmetric torsion, that is, one in which the three Bismut connections asso-ciated with the three Hermitian structures (Jr , h) coincide for r = 1, 2, 3. Thisgeometry was introduced by Howe and Papadopoulos [1996] and later studied in[Grantcharov and Poon 2000; Fino and Grantcharov 2004; Barberis et al. 2009;Barberis and Fino 2008; Swann 2010].
In the interesting special case in which the torsion 3-form of such a hyper-Hermitian connection is closed, the HKT manifold is called strong.
In Section 7, Theorem 7.1 gives conditions under which an S1-bundle over a(4n+3)-dimensional manifold endowed with three almost contact metric struc-tures is HKT and in particular when it is strong HKT.
2. SKT structures arising from S1-bundles
Consider a (2n+1)-manifold N 2n+1 endowed with an almost contact metric struc-ture (I, ξ, η, g); that is, I is a tensor field of type (1, 1), ξ is a vector field, η is a1-form, and g is a Riemannian metric on N 2n+1, satisfying together the conditions
I 2=− Id+η⊗ ξ, η(ξ)= 1, g(IU, I V )= g(U, V )− η(U )η(V )
for any vector fields U and V on N 2n+1. Denote by ω the fundamental 2-form of(I, ξ, η, g); that is, ω is the 2-form on N 2n+1 given by
ω( · , · )= g( · , I · ).
Given the tensor field I , consider its Nijenhuis torsion [I, I ], defined by
(1) [I, I ](X, Y )= I 2[X, Y ] + [I X, I Y ] − I [I X, Y ] − I [X, I Y ].
On the product N 2n+1×R, one can define a natural almost complex structure
J(
X, f ddt
)=
(I X + f ξ, −η(X) d
dt
),
where f is a C∞-function on N 2n+1×R and t is the coordinate on R.
52 MARISA FERNÁNDEZ, ANNA FINO, LUIS UGARTE AND RAQUEL VILLACAMPA
Definition 2.1 [Sasaki and Hatakeyama 1961]. We call an almost contact met-ric structure (I, ξ, η, g) on N 2n+1 normal if the almost complex structure J onN 2n+1
×R is integrable, or equivalently if
[I, I ](X, Y )+ 2dη(X, Y )ξ = 0
for any vector fields X, Y on N 2n+1.
By [Blair 1967, Lemma 2.1], one has iξdη= 0 for a normal almost contact metricstructure (I, ξ, η, g).
Remark 2.2. The normality of the almost contact structure implies as well thatI dη= dη. Indeed, d(η− i dt)= dη has no (0, 2)-part and therefore has no (2, 0)-part since dη is real. Thus, J dη = dη, but Jdη = I dη as well since iξdη = 0.
We recall that a Hermitian manifold (M, J, h) is SKT if and only if the 3-formJ dF is closed, where F is the fundamental 2-form of (J, h). We will use theconvention that J acts on r -forms β by
(Jβ)(X1, . . . , Xr )= β(J X1, . . . , J Xr ) for any vector fields X1, . . . , Xr .
We now show general conditions under which an S1-bundle over an almostcontact metric (2n+1)-dimensional manifold is SKT.
Let (N 2n+1, I, ξ, η) be a (2n+1)-dimensional almost contact manifold, andlet � be a closed 2-form on N 2n+1 that represents an integral cohomology classon N 2n+1. From the well-known result of Kobayashi [1956], we can consider thecircle bundle S1 ↪→ P→ N 2n+1 and the connection 1-form θ on P whose curvatureform is dθ = π∗(�), where π : P→ N 2n+1 is the projection.
By using the almost contact structure (I, ξ, η) and the connection 1-form θ , onecan define an almost complex structure J on P as follows [Ogawa 1963]. For anyright-invariant vector field X on P, the vector field J X is given by
(2) θ(J X)=−π∗(η(π∗X)) and π∗(J X)= I (π∗X)+ θ (X)ξ,
where θ (X) is the unique function on N 2n+1 such that
(3) π∗θ (X)= θ(X).
This definition can be extended to an arbitrary vector field X on P since X canbe written in the form X =
∑j f j X j , with f j smooth functions on P , and X j
right-invariant vector fields. Then J X =∑
j f j J X j .Ogawa [1963] showed that when (N 2n+1, I, ξ, η) is normal, the almost complex
structure J on P defined by (2) is integrable if and only if dθ is J -invariant, thatis, if J (dθ)= dθ or equivalently dθ(J X, Y )+dθ(X, JY )= 0 for any vector fieldsX and Y on P . That is, dθ is a complex 2-form on P having bidegree (1, 1) withrespect to J .
STRONG KÄHLER WITH TORSION STRUCTURES 53
In terms of the 2-form �, whose lift to P is the curvature of the circle bundleS1 ↪→ P → N 2n+1, the previous condition means that � is I -invariant, that is,I (�)=�. Therefore iξ�= 0.
If {e1, . . . , e2n, η} is an adapted coframe on a neighborhood U on N 2n+1, thatis,
I e2 j−1=−e2 j and I e2 j
= e2 j−1 for 1≤ j ≤ n,
then we can take {π∗e1, . . . , π∗e2n, π∗η, θ} as a coframe in π−1(U ). By usingthe coframe {π∗e1, . . . , π∗e2n
}, we may write
dθ = π∗α+π∗β ∧π∗η,
where α is a 2-form in∧2〈e1, . . . , e2n
〉, and β ∈∧1〈e1, . . . , e2n
〉.Suppose that N 2n+1 has a normal almost contact metric structure (I, ξ, η, g). We
consider a principal S1-bundle P with base space N 2n+1 and connection 1-form θ ,and endow P with the almost complex structure J (associated to θ ) defined by (2).Since N 2n+1 has a Riemannian metric g, a Riemannian metric h on P compatiblewith J (see [Ogawa 1963]) is given by
(4) h(X, Y )= π∗g(π∗X, π∗Y )+ θ(X) θ(Y )
for any right-invariant vector fields X and Y . This definition can be extended toany vector field on P .
Theorem 2.3. Consider a (2n+1)-dimensional almost contact metric manifold(N 2n+1, I, ξ, η, g), and let � be a closed 2-form on N 2n+1 that represents anintegral cohomology class. Consider the circle bundle S1 ↪→ P → N 2n+1 withconnection 1-form θ , whose curvature form is dθ = π∗(�) for the projectionπ : P→ N 2n+1.
The almost Hermitian structure (J, h) on P defined by (2) and (4) is SKT if andonly if (I, ξ, η, g) is normal, dθ is J -invariant, and
(5)d(π∗(I (iξdω)))= 0,
d(π∗(I (dω)− dη∧ η))= (−π∗(I (iξdω))+π∗�)∧π∗�,
where ω is the fundamental form of the almost contact metric structure (I, ξ, η, g).
Proof. As we mentioned, a result of Ogawa [1963] asserts that the almost complexstructure J is integrable if and only if (I, ξ, η, g) is normal and J (dθ)= dθ . Thus,(J, h) is SKT if and only if the 3-form J dF is closed. Using the first equalityin (2), we find that the fundamental 2-form F on P is
F(X, Y )= h(X, JY )
= π∗g(π∗X, π∗ JY )+ θ(X) θ(JY )
= π∗g(π∗X, π∗ JY )− θ(X) π∗η(π∗Y ).
54 MARISA FERNÁNDEZ, ANNA FINO, LUIS UGARTE AND RAQUEL VILLACAMPA
Therefore, taking into account that we are working with a circle bundle (whosefiber is thus 1-dimensional), we have
(6)
F = π∗ω+π∗η∧ θ,
d F = π∗(dω)+π∗(dη)∧ θ −π∗η∧ dθ,
J dF = J (π∗(dω))− J (π∗(dη))∧π∗η− θ ∧ dθ
since J (π∗η)= θ and J is integrable, and so J (dθ)= dθ . Moreover,
(7) J (π∗(dω))= π∗(I (dω))+π∗(I (iξdω))∧ θ.
Indeed, locally and in terms of the adapted basis {e1, . . . , e2n+1} with
I e2 j−1=−e2 j for 1≤ j ≤ n, I e2n+1
= 0, and η = e2n+1,
we can write dω = α + β ∧ η, where the local forms α ∈∧3〈e1, . . . , e2n
〉 andβ ∈
∧2〈e1, . . . , e2n
〉 are generated only by e1, . . . , e2n . Furthermore, we haveIα = I (dω) and β = iξdω. Thus
J (π∗(dω))= J (π∗(α))+ J (π∗(iξdω))∧ θ.
Now, by using (2) and (3), we see that J (π∗(α)) = π∗(Iα) and J (π∗(iξdω)) =π∗(I (iξdω)), which proves (7). As a consequence of Remark 2.2,
(8) J (π∗(dη))= π∗(I (dη))−π∗(I (iξdη))∧ θ = π∗(dη)
since iξdη = 0 and I dη = dη.Using (7) and (8), we get
(9) J dF = π∗(I (dω))+π∗(I (iξdω))∧ θ −π∗(dη)∧π∗η− θ ∧ dθ.
Therefore,
d(J dF)= d(π∗(I (dω)))+ d(π∗(I (iξdω)))∧ θ +π∗(I (iξdω))∧ dθ
− d(π∗(dη))∧π∗η−π∗(dη)∧ dπ∗η− dθ ∧ dθ.
Consequently, d(J dF)= 0 if and only if
d(π∗(I (iξdω)))= 0,
d(π∗(I (dω)− dη∧ η))= (π∗(−I (iξdω))+ dθ)∧ dθ. �
An almost contact metric manifold (N 2n+1, I, ξ, η, g) is quasi-Sasakian if it isnormal and its fundamental form ω is closed. In particular, if dη = α ω, then thealmost contact metric structure is called α-Sasakian. When α =−2, the structureis said to be Sasakian.
By [Friedrich and Ivanov 2002, Theorem 8.2], an almost contact metric manifold(N 2n+1, I, ξ, η, g) admits a connection ∇c that preserves the almost contact metric
STRONG KÄHLER WITH TORSION STRUCTURES 55
structure and has totally skew-symmetric torsion tensor if and only if the Nijenhuistensor of I , given by (1), is skew-symmetric and ξ is a Killing vector field. Thisconnection is unique.
In particular, on any quasi-Sasakian manifold (N 2n+1, I, ξ, η, g) there exists aunique connection ∇c with totally skew-symmetric torsion, such that
∇c I = 0, ∇cg = 0, ∇cη = 0.
Such a connection ∇c is uniquely determined by
(10) g(∇cX Y, Z)= g(∇g
X Y, Z)+ 12(dη∧ η)(X, Y, Z),
where ∇g is the Levi-Civita connection and 12(dη∧η) is the torsion 3-form of ∇c.
Corollary 2.4. Let (N 2n+1, I, ξ, η, g) be a quasi-Sasakian (2n+1)-manifold, andlet � be a closed 2-form on N 2n+1 that represents an integral cohomology class.Consider the circle bundle S1 ↪→ P → N 2n+1 with connection 1-form θ whosecurvature form is dθ = π∗(�) for the projection π : P → N 2n+1. The almostHermitian structure (J, h) on P defined by (2) and (4) is SKT if and only if � isI -invariant, iξ�= 0, and
(11) dη∧ dη =−�∧�.
The Bismut connection ∇B of (J, h) on P and the connection ∇c on N given by(10) are related by
(12) h(∇BX Y, Z)= π∗g(∇c
π∗Xπ∗Y, π∗Z)
for any vector fields X, Y, Z ∈ Ker θ .
Proof. Since dω = 0, if we impose the SKT condition, then we get by using theprevious theorem the equation (11).
The Bismut connection ∇B associated to the Hermitian structure (J, h) on P is
(13) h(∇BX Y, Z)= h(∇h
X Y, Z)− 12 dF(J X, JY, J Z)
for any vector fields X, Y, Z on P , where ∇h is the Levi-Civita connection asso-ciated to h. Then, for any X, Y, Z in the kernel of θ , we have
h(∇BX Y, Z)= π∗g(∇h
X Y, Z)+ 12(π∗(dη)∧π∗η)(X, Y, Z).
By [Ogawa 1963, Lemma 3] and the definition of ∇c, we get
h(∇BX Y, Z)= π∗g(∇g
π∗Xπ∗Y, π∗Z)+12(π∗(dη)∧π∗η)(X, Y, Z)
= π∗g(∇cπ∗Xπ∗Y, π∗Z)
for any X, Y, Z in the kernel of θ . �
56 MARISA FERNÁNDEZ, ANNA FINO, LUIS UGARTE AND RAQUEL VILLACAMPA
Remark 2.5. If the structure (I, ξ, η, g) is α-Sasakian, equation (11) reads
�∧�=−α2ω∧ω.
In the case of a trivial S1-bundle, that is, if we consider the natural almostHermitian structure on the product N 2n+1
×R, we get this:
Corollary 2.6. Let (N 2n+1, I, ξ, η, g) be a (2n+1)-dimensional almost contactmetric manifold. Impose on the product N 2n+1
×R the almost complex structure Jgiven by
J X = I X for X ∈ Ker η and Jξ =− ddt,
and the metric h given by h = g + (dt)2. The Hermitian structure (J, h) is SKTif and only if (I, ξ, η, g) is normal, d(I (dω)) = d(dη ∧ η) and d(I (iξdω)) = 0,where ω denotes the fundamental 2-form of the almost contact metric structure(g, I, ξ, η).
Corollary 2.7. Let (N 2n+1, I, ξ, η, g) be a (2n+1)-dimensional quasi-Sasakianmanifold with dη∧ dη = 0. The Hermitian structure (J, h) on N 2n+1
×R is SKT.Moreover, its Bismut connection ∇B coincides with the unique connection ∇c onN 2n+1 given by (10).
Proof. In this case, since dω = 0 we get d(J dF) = −d(dη ∧ η). By using (12),we get h(∇B
X Y, Z)= g(∇cX Y, Z) for any vector fields X, Y, Z on N 2n+1. �
2.1. Examples. We will present three examples of quasi-Sasakian Lie algebrassatisfying the condition dη∧dη= 0. By applying Corollary 2.7, one gets an SKTstructure on the product of the corresponding simply connected Lie group by R.
Example 2.8. Let s be the 5-dimensional Lie algebra with structure equations
de1= e13
+ e23+ e25
− e34+ e35,
de2= 2e12
− 2e13+ e14
− e15− e24
+ e34+ e45,
de3=−e12
+ e13+ e14
− e15+ 2e24
− 2e34+ e45,
de4=−e12
− e23+ e24
− e25− e35,
de5= e12
− e13− e24
+ e34,
where ei j= ei∧ e j . On s, take the quasi-Sasakian structure (I, ξ, η, g) given by
(14) η = e5, I e1=−e2, I e3
=−e4, ω =−e12− e34, g =
∑5j=1(e
j )2.
This quasi-Sasakian structure satisfies the condition d(dη ∧ η) = 0. The Liealgebra s is 2-step solvable since the commutator
s1= [s, s] = R〈e1− e4, e2+ e3, e1− e2+ 2e3− e5〉
STRONG KÄHLER WITH TORSION STRUCTURES 57
is abelian, where {e1, . . . , e5} denotes the dual basis of {e1, . . . , e5}. Moreover, s
has trivial center, is irreducible and nonunimodular, since the trace of ade1 is −3.
Example 2.9. Consider the family of 2-step solvable Lie algebras sa for a∈R−{0},given by
de1= a e23
+ 3e25, de3= a e34,
de2=−a e13
− 3e15, de4= 0,
de5=−
13a2 e34.
The almost contact metric structure (I, ξ, η, g) defined in (14) is quasi-Sasakianand satisfies the condition dη ∧ dη = 0. The second cohomology group of sa isgenerated by e12 and e45.
Example 2.10. Another family of quasi-Sasakian Lie algebras that satisfies thecondition dη∧ dη = 0 is gb for b ∈ R−{0}, with structure equations
de1= b(e13
+ e14− e23
+ e24)+ e25, de3= 2e45,
de2= b(−e13
+ e14− e23
− e24)− e15, de4=−2e35,
de5=−4b2e34,
and endowed with the quasi-Sasakian structure given by (14). The second coho-mology group of gb is generated by e12. The Lie algebras gb are not solvable sincethe commutators are [gb, gb] = gb.
The Lie groups underlying Examples 2.9 and 2.10 also satisfy the conditions ofCorollary 2.4 with �∧� = 0, by just taking as connection 1-form the 1-form e6
such that de6= λe12. Then, �= λe12. With this expression of de6, we have
d2e6= 0, J (de6)= de6, and de6
∧ de6= 0.
Therefore, equation (11) is satisfied. Observe that λ = 0 provides examples oftrivial S1-bundles.
The next example allows us to recover one of the 6-dimensional nilmanifoldsfound in [Fino et al. 2004]:
Example 2.11. Consider the 5-dimensional nilpotent Lie algebra with structureequations
de j= 0 for j = 1, . . . , 4,
de5= e12
+ e34,
and endowed with the quasi-Sasakian structure given by (14). If we consider theclosed 2-form � = e13
+ e24 and apply Corollary 2.4, we see that there existsa nontrivial S1-bundle over the corresponding 5-dimensional nilmanifold. Sincede5∧de5
=−�∧� 6= 0, the total space of this S1-bundle is an SKT nilmanifold.More precisely, according to the classification given in [Fino et al. 2004] (see also
58 MARISA FERNÁNDEZ, ANNA FINO, LUIS UGARTE AND RAQUEL VILLACAMPA
[Ugarte 2007]), the nilmanifold is the one with underlying Lie algebra isomorphicto h3⊕h3, where by h3 we denote the real 3-dimensional Heisenberg Lie algebra.
Since the starting Lie algebra from Example 2.11 is Sasakian, it is natural to startwith other 5-dimensional Sasakian Lie algebras to construct new SKT structures indimension 6. A classification of 5-dimensional Sasakian Lie algebras was obtainedin [Andrada et al. 2009].
Example 2.12. Consider the 5-dimensional Lie algebra k3 with structure equations
de1= 0, de4
= 0,
de2=−e13, de5
= λe14+µe23,
de3= e12,
where λ,µ<0. By [Andrada et al. 2009], this algebra admits the Sasakian structuregiven by
I e1= e4, I e2
= e3, η = e5,
g =− 12λ(e1)
2−
12λ(e2)
2−
12µ(e3)
2−
12µ(e4)
2+ (e5)
2,
and is isomorphic to R n (h3 ×R); moreover, the corresponding solvable simplyconnected Lie group admits a compact quotient by a discrete subgroup.
Consider on k3 the closed 2-form�=λe14−µe23. The form� is I -invariant and
satisfies�∧�=−2λµe1234. Since e5 is the contact form and de5∧de5
=2λµe1234,we get again by Corollary 2.4 an SKT structure on a nontrivial S1-bundle over the5-dimensional solvmanifold. We denote by e6 the connection 1-form.
The orthonormal basis {α1= e1, α2
= e4, α3= e2, α4
= e3, α5= e5, α6
= θ}
for the SKT metric satisfies the equations
dα1= dα2
= 0, dα3=−α14, dα4
= α13,
dα5= λα12
+µα34, dα6= λα12
−µα34,
and the complex structure is given by J (X1)= X2, J (X3)= X4 and J (X5)= X6,where {X i }
6i=1 denotes the basis dual to {αi
}6i=1. Since the fundamental 2-form is
F = α12+α34
+α56, the 3-form torsion T of the SKT structure is
T = λα12(α5+α6)+µα34(α5
−α6).
Moreover, ∗T = λα12(α5+ α6)− µα34(α5
− α6), where ∗ denotes the metric’sHodge operator; this implies that the torsion form is also coclosed.
The only nonzero curvature forms (�B)ij of the Bismut connection ∇B are
(�B)12 =−2λ2α12 and (�B)34 =−2µ2α34.
STRONG KÄHLER WITH TORSION STRUCTURES 59
A direct calculation shows that the 1-forms α5 and α6 and the 2-forms α12 and α34
are parallel with respect to the Bismut connection, which implies that ∇B T = 0.Finally, Hol(∇B)=U (1)×U (1)⊂U (3) since ∇Bαi
6= 0 for i = 1, 2, 3, 4.
3. SKT structures arising from Riemannian cones
Let N 2n+1 be a (2n+1)-dimensional manifold endowed with an almost contactmetric structure (I, ξ, η, g), and denote by ω its fundamental 2-form.
The Riemannian cone of N 2n+1 is defined as the manifold N 2n+1×R+ equipped
with the cone metric
(15) h = t2g+ (dt)2.
The cone N 2n+1×R+ has a natural almost Hermitian structure defined by
(16) F = t2ω+ tη∧ dt.
The almost complex structure J on N 2n+1×R+ defined by (F, h) is given by
J X = I X for X ∈ Ker η and Jξ =−t ddt.
In terms of a local orthonormal adapted coframe {e1, . . . , e2n} for g with
(17) ω =−
n∑j=1
e2 j−1∧ e2 j ,
we have
(18)Je2 j−1
=−e2 j , Je2 j= e2 j−1 for j = 1, . . . , n,
J (te2n+1)= dt, J (dt)=−te2n+1.
The almost Hermitian structure (J, h) on N 2n+1×R+ is Kahler if and only if the
almost contact metric structure (I, ξ, η, g) on N 2n+1 is Sasakian, that is, a normalcontact metric structure.
If we impose that the almost Hermitian structure (J, h) on N 2n+1×R+ is SKT,
we can prove the following:
Theorem 3.1. Consider a (2n+1)-dimensional almost contact metric manifold(N 2n+1, I, ξ, η, g). The almost Hermitian structure (J, h) given by (15) and (16)on the Riemannian cone (N 2n+1
×R+, h) is SKT if and only if (I, ξ, η, g) is normaland
(19) −4η∧ω+ 2 I (dω)− 2dη∧ η = d(I (iξdω)),
where ω denotes the fundamental 2-form of the almost contact metric structure(I, ξ, η, g).
60 MARISA FERNÁNDEZ, ANNA FINO, LUIS UGARTE AND RAQUEL VILLACAMPA
Proof. J is integrable if and only if the almost contact metric structure is normal.We compute J dF . We have
d F = 2tdt ∧ω+ t2dω+ t dη∧ dt, and so
JdF =−2t2η∧ω+ t2 J (dω)− t2dη∧ η
since Jω=ω, J (dt)=−tη and J dη= dη. Moreover, with respect to an adaptedbasis {e1, . . . , e2n+1
} we can get, in a way similar to the proof of Theorem 2.3, that
(20) J dω = I (dω)+ I (iξdω)∧ Jη.
As a consequence, we get J dF =−2t2η∧ω+t2 I (dω)+t dt∧ I (iξdω)−t2 dη∧η.Therefore, by imposing that d(J dF)= 0, we obtain
−4η∧ω+ 2I (dω)− 2dη∧ η− d(I (iξdω))= 0,
−2d(η∧ω)+ d(I (dω))− d(dη∧ η)= 0.
Since the second equation is a consequence of the first, the Hermitian structure(F, h) on the Riemannian cone N 2n+1
× R+ is SKT if and only if the almostcontact metric structure (I, η, ξ, g, ω) on N 2n+1 satisfies equation (19). �
Remark 3.2. As a consequence of previous theorem, when n = 1, equation (19)is satisfied if and only if the 3-dimensional manifold N is Sasakian. On the otherhand, if n > 1 and the almost contact metric structure on N 2n+1 is quasi-Sasakian(that is, dω = 0), then the structure has to be Sasakian, that is, dη =−2ω.
Example 3.3. Consider the 5-dimensional Lie algebras ga,b,c with structure equa-tions
de1= a e23
+ 2e25+
(−
12ab+ b3
2a+ 2b
a
)e34+ be45,
de2=−a e13
− 2e15−
12 bce34
− be35,
de3=
(−
4a−
b2
a
)e34,
de4= ce34,
de5= 2e12
+ be14− be23
+ (2+ b2)e34,
where a, b, c ∈ R and a 6= 0. They are endowed with the normal almost contactmetric structure (I, ξ, η, g, ω) with
I e1=−e2, I e3
=−e4, η = e5, ω =−e12− e34.
This structure satisfies (19), and therefore the Riemannian cones over the corre-sponding simply connected Lie groups are SKT.
STRONG KÄHLER WITH TORSION STRUCTURES 61
4. SKT SU(3)-structures
Let (M6, J, h) be a 6-dimensional almost Hermitian manifold. An SU(3)-structureon M6 is determined by the choice of a (3, 0)-form 9 = 9++ i9− of unit norm.If 9 is closed, then the underlying almost complex structure J is integrable andthe manifold is Hermitian. We will denote the SU(3)-structure (J, h, 9) simply by(F, 9), where F is the fundamental 2-form, since from F and9 we can reconstructthe almost Hermitian structure.
Definition 4.1. We say that an SU(3)-structure (F, 9) on M6 is SKT if
(21) d9 = 0 and d(J dF)= 0,
where J is the associated complex structure.
We will see the relation between SKT SU(3)-structures in dimension 6 andSU(2)-structures in dimension 5.
First, we recall some facts about SU(2)-structures on a 5-dimensional mani-fold. An SU(2)-structure on a 5-dimensional manifold N 5 is an SU(2)-reductionof the principal bundle of linear frames on N 5. By [Conti and Salamon 2007,Proposition 1], these structures are in one-to-one correspondence with quadru-plets (η, ω1, ω2, ω3), where η is a 1-form and ωi are 2-forms on N 5 satisfyingωi ∧ω j = δi jv and v∧η 6= 0 for some 4-form v, and ω2(X, Y )≥ 0 if iX ω3= iY ω1,where iX denotes the contraction by X . Equivalently, an SU(2)-structure on N 5
can be viewed as the datum of (η, ω1,8), where η is a 1-form, ω1 is a 2-form, and8= ω2+ i ω3 is a complex 2-form such that
η∧ω1 ∧ω1 6= 0, 8∧8= 0, ω1 ∧8= 0, 8∧8= 2ω1 ∧ω1,
and 8 is of type (2, 0) with respect to ω1.Conti and Salamon [2007] locally characterize an SU(2)-structure as follows. If
(η, ω1, ω2, ω3) is an SU(2)-structure on a 5-manifold N 5, then locally there existsan orthonormal basis of 1-forms {e1, . . . , e5
} such that
ω1 = e12+ e34, ω2 = e13
− e24, ω3 = e14+ e23, η = e5 .
We can also consider the local tensor field I given by
I e1=−e2, I e2
= e1, I e3=−e4, I e4
= e3, I e5= 0.
This tensor gives rise to a global tensor field of type (1, 1) on the manifold N 5,defined by ω1(X, Y )= g(X, I Y ) for any vector fields X and Y on N 5, where g isthe Riemannian metric on N 5 underlying the SU(2)-structure. The tensor field Isatisfies I 2
=− Id+η⊗ ξ , where ξ is the vector field on N 5 dual to the 1-form η.Therefore, given an SU(2)-structure (η, ω1, ω2, ω3)we also have an almost con-
tact metric structure (I, ξ, η, g) on the manifold, where ω1 is its fundamental form.
62 MARISA FERNÁNDEZ, ANNA FINO, LUIS UGARTE AND RAQUEL VILLACAMPA
Remark 4.2. Notice that we have two more almost contact metric structures whenwe consider ω2 and ω3 as fundamental forms.
If N 5 has an SU(2)-structure (η, ω1, ω2, ω3), the product N 5×R has a natural
SU(3)-structure given by
(22) F = ω1+ η∧ dt and 9 = (ω2+ iω3)∧ (η− i dt).
By Corollary 2.6, the previous SU(3)-structure is SKT if and only if
(23)d(I (dω1))= d(dη∧ η), dω2 =−3ω3 ∧ η,
d(I (iξ dω1))= 0, dω3 = 3ω2 ∧ η,
proving this:
Theorem 4.3. Suppose N 5 is a 5-dimensional manifold endowed with an SU(2)-structure (η, ω1, ω2, ω3). The SU(3)-structure (F, 9) given by (22) on the productN 5×R is SKT if and only if the equations (23) are satisfied.
Example 4.4. On the 5-dimensional Lie algebras introduced in Examples 2.8, 2.9and 2.10, consider the SU(2)-structure given by
ω = ω1 = e12+ e34, ω2 = e13
− e24, ω3 = e14+ e23.
For Example 2.8, we have
dω2 =−2ω3 ∧ η− 4(e124− e134) and dω3 = 2ω2 ∧ η+ 4(e123
+ e234).
For Examples 2.9 and 2.10, we get dω2 = −3ω3 ∧ η and dω3 = 3ω2 ∧ η.
Therefore one gets an SKT SU(3)-structure on the product of the correspondingsimply connected Lie groups by R.
We will study the existence of SKT SU(3)-structures on a Riemannian cone overa 5-dimensional manifold N 5 endowed with an SU(2)-structure (η, ω1, ω2, ω3).Then N 5 has an induced almost contact metric structure (I, ξ, η, g), and ω1 is itsfundamental form.
The Riemannian cone (N 5× R+, h) of (N 5, g) has a natural SU(3)-structure
defined by
F = t2ω1+ tη∧ dt and 9 = t2(ω2+ iω3)∧ (tη− idt).
In terms of a local orthonormal coframe {e1, . . . , e5} for g such that
ω1 =−e12− e34, ω2 =−e13
+ e24, ω3 =−e14− e23, η = e5,
we haveJe1=−e2, Je2
= e1, Je3=−e4,
Je4= e3, J (te5)= dt, J (dt)=−te5.
STRONG KÄHLER WITH TORSION STRUCTURES 63
We recall that the SU(3)-structure (F, 9) on N 5×R+ is integrable if and only
if the SU(2)-structure (η, ω1, ω2, ω3) on N 5 is Sasaki–Einstein, or equivalently ifand only if
dη =−2ω1, dω2 =−3ω3 ∧ η, dω3 = 3ω2 ∧ η.
For the Riemannian cones, we can prove the following
Corollary 4.5. Let N 5 be a 5-dimensional manifold endowed with an SU(2)-structure (η, ω1, ω2, ω3). The SU(3)-structure (F, 9) on the Riemannian cone(N 5×R+, h) is SKT if and only if
(24)
−4η∧ω1+ 2I (dω1)− 2dη∧ η = d(I (iξdω1)),
dω2 = 3ω3 ∧ η,
dω3 =−3ω2 ∧ η.
Proof. By imposing that d9 = 0, we get the conditions dω2 = −3ω3 ∧ η anddω3= 3ω2∧η. By imposing d(J dF)= 0, we get, as in the proof of Theorem 3.1,equation (19) for ω = ω1. �
5. Almost contact metric structure induced on a hypersurface
We study the almost contact metric structure induced naturally on any orientedhypersurface N 2n+1 of a (2n+2)-manifold M2n+2 equipped with an SKT structure.
Let f : N 2n+1→ M2n+2 be an oriented hypersurface of a (2n+2)-dimensional
manifold M2n+2 endowed with an SKT structure (J, h, F), and denote by U theunitary normal vector field. It is well known that N 2n+1 inherits an almost contactmetric structure (I, ξ, η, g) such that η and the fundamental 2-form ω are given by
(25) η =− f ∗(iU F) and ω = f ∗F,
where F is the fundamental 2-form of the almost Hermitian structure; see, forinstance, [Blair 2002].
Proposition 5.1. Suppose f : N 2n+1→ M2n+2 is an immersion of an oriented
(2n+1)-dimensional manifold into a (2n+2)-dimensional Hermitian manifold. Ifthe Hermitian structure (J, h) on M2n+2 is SKT , then the induced almost contactmetric structure (I, ξ, η, g) on N 2n+1, with η and ω given by (25), satisfies
(26) d(I dω− I ( f ∗(iU d F))∧ η)= 0.
Proof. Locally we can choose an adapted coframe {e1, . . . , e2n+2} for the Hermit-
ian structure so that the unitary normal vector field U is dual to e2n+2. Since the
64 MARISA FERNÁNDEZ, ANNA FINO, LUIS UGARTE AND RAQUEL VILLACAMPA
almost complex structure J is given in this adapted basis by
Je2 j−1=−e2 j , Je2 j
= e2 j−1 for j= 1, . . . , n,
Je2n+1= e2n+2, Je2n+2
=−e2n+1,
it follows that the tensor field I on N 2n+1 has I f ∗ei= f ∗ Jei for i = 1, . . . , 2n+1.
That is,
I f ∗e2 j−1=− f ∗e2 j , I f ∗e2 j
= f ∗e2 j−1 for j = 1, . . . , n, I f ∗e2n+1= 0.
However, I f ∗e2n+2= 0 6= f ∗e2n+1
=− f ∗ Je2n+2.We compute f ∗ J dF . First, we decompose (locally and in terms of the adapted
basis) the differential of F as
dF = α+β ∧ e2n+1+ γ ∧ e2n+2
+µ∧ e2n+1∧ e2n+2,
where the local forms
α ∈∧3〈e1, . . . , e2n
〉, β, γ ∈∧2〈e1, . . . , e2n
〉, µ ∈∧1〈e1, . . . , e2n
〉
are generated only by e1, . . . , e2n . Then,
J dF = Jα+ Jβ ∧ e2n+2− Jγ ∧ e2n+1
+ Jµ∧ e2n+1∧ e2n+2.
Since f ∗e2n+2= 0 and f ∗e2n+1
= η, we have f ∗ J dF = f ∗ Jα − ( f ∗ Jγ ) ∧ η.However, f ∗(iU dF)= f ∗γ + f ∗µ∧ η, which implies that
I ( f ∗(iU dF))= I f ∗γ = f ∗ Jγ.
On the other hand, I dω = I d f ∗F = I f ∗dF = I f ∗α = f ∗ Jα. We conclude that
f ∗ J dF = f ∗ Jα− ( f ∗ Jγ )∧ η = I dω− I ( f ∗(iU d F))∧ η.
Now, if the Hermitian structure is SKT, then JdF is closed and the induced struc-ture satisfies (26). �
Remark 5.2. Notice that, using iU d F = LU F − diU F , we can write (26) as
d(I dω− I ( f ∗(LU F)+ dη)∧ η)= 0.
Therefore, if f ∗(LU F)= 0, then the induced almost contact metric structure has tosatisfy the equation d(I dω− I (dη)∧η)= 0. In the case of the product N 2n+1
×R,the condition f ∗(LU F) = 0 is satisfied. In the case of the Riemannian cone, wehave Ld/dt F = 2tω + dt ∧ η and therefore f ∗(Ld/dt F) = 2ω. In this way, werecover some of the equations obtained in Corollary 2.6 and Theorem 3.1.
Now we study the structure that is induced naturally on any oriented hypersur-face N 5 of a 6-manifold M6 equipped with an SKT SU(3)-structure.
STRONG KÄHLER WITH TORSION STRUCTURES 65
Let f : N 5→ M6 be an oriented hypersurface of a 6-manifold M6 endowed
with an SU(3)-structure (F, 9 =9++ i 9−), and denote by U the unitary normalvector field. Then N 5 inherits an SU(2)-structure (η, ω1, ω2, ω3) given by
(27) η =− f ∗(iU F), ω1 = f ∗F, ω2 =− f ∗(iU9−), ω3 = f ∗(iU9+).
Corollary 5.3. Let f : N 5→ M6 be an immersion of an oriented 5-dimensional
manifold into a 6-dimensional manifold with an SU(3)-structure. If the SU(3)-structure is SKT , then the induced SU(2)-structure on N 5 given by (27) satisfies
d(I dω1− I f ∗(iU d F)∧ η)= 0,(28)
d(ω2 ∧ η)= 0 and d(ω3 ∧ η)= 0.(29)
Proof. Equation (28) follows from Proposition 5.1 by taking ω = ω1. Locally,we can choose an adapted coframe {e1, . . . , e5, e6
} for the SU(3)-structure suchthat the unitary normal vector field U is dual to e6. From (27), it follows thatω2∧η= f ∗9+ and ω3∧η= f ∗9−. If 9 =9++ i 9− is closed, then the inducedstructure satisfies (29). �
5.1. A simple example. Consider the 6-dimensional nilmanifold M6 whose un-derlying nilpotent Lie algebra has structure equations
de j= 0 for j = 1, 2, 3, 6, de4
= e12, de5= e14,
and is endowed with the SU(3)-structure given by
F =−e14− e26
− e53 and 9 = (e1− ie4)∧ (e2
− ie6)∧ (e5− ie3).
The oriented hypersurface with normal vector field dual to e2 is a 5-dimensionalnilmanifold N 5 that by [Conti and Salamon 2007] has no invariant hypo structures.However, the SU(2)-structure on N 5, namely,
(30) η = e2, ω1 =−e14− e53, ω2 =−e15
− e34, ω3 =−e13− e45,
satisfies (28) and (29). In Section 6, we will show that by using this SU(2)-structureand appropriate evolution equations, we can construct an SKT SU(3)-structure onthe product of N 5 with an open interval.
6. SKT evolution equations
The goal here is to construct SKT SU(3)-structures by using appropriate evolutionequations, starting from a suitable SU(2)-structure on a 5-dimensional manifold.We follow ideas of [Hitchin 2001] and [Conti and Salamon 2007].
Lemma 6.1. Let (η(t), ω1(t), ω2(t), ω3(t)) be a family of SU(2)-structures on a5-dimensional manifold N 5 for t ∈ (a, b). The SU(3)-structure on M6
= N 5×(a, b)
66 MARISA FERNÁNDEZ, ANNA FINO, LUIS UGARTE AND RAQUEL VILLACAMPA
given by F =ω1(t)+η(t)∧dt and9 = (ω2(t)+iω3(t))∧(η(t)−i dt
)satisfies the
condition d9 = 0 if and only if (η(t), ω1(t), ω2(t), ω3(t)) is an SU(2)-structuresuch that, for any t in the open interval (a, b),
(31)d(ω2(t)∧ η(t))= 0, ∂t(ω2(t)∧ η(t))=−dω3(t),
d(ω3(t)∧ η(t))= 0, ∂t(ω3(t)∧ η(t))= dω2(t).
Here, d denotes the exterior differential on N 5 and d is the exterior differentialon M6. We now present the additional evolution equations to be added to the lasttwo of (31) in order to ensure that d Jd F = 0.
Proposition 6.2. Let (η(t), ω1(t), ω2(t), ω3(t)) be a family of SU(2)-structures onN 5 for t ∈ (a, b). The SU(3)-structure on M6
= N 5× (a, b) given by
(32) F = ω1(t)+ η(t)∧ dt and 9 = (ω2(t)+ iω3(t))∧ (η(t)− i dt),
has J dF closed if and only if (η(t), ω1(t), ω2(t), ω3(t)) satisfies the evolutionequations
(33)
d(It dω1(t)− It(∂tω1(t)+ dη(t))∧ η(t)
)= 0,
∂t(It dω1(t)− It(∂tω1(t)+ dη(t))∧ η(t)
)=−d
(It(iξ dω1(t))− It(iξ (∂tω1(t)+ dη(t)))∧ η(t)
),
where ξ(t) denotes the vector field on N 5 dual to η(t) for each t ∈ (a, b).
Proof. Since F = ω1(t) + η(t) ∧ dt , we have dF = dω1 + (∂tω1 + dη) ∧ dt .Define {e1(t), . . . , e4(t), η(t)} to be a local adapted basis for the SU(2)-structure(η(t), ω1(t), ω2(t), ω3(t)). Then, {e1(t), . . . , e4(t), η(t), dt} is an adapted basisfor the SU(3)-structure (32), and J is given by
Je1(t)=−e2(t), Je2(t)= e1(t), Jη(t)= dt,
Je3(t)=−e4(t), Je4(t)= e3(t), J dt =−η(t).
For each t , the structures It induced on N 5 are given by
It e1(t)=−e2(t), It e2(t)= e1(t),
It e3(t)=−e4(t), It e4(t)= e3(t), Itη(t)= 0.
Now, we can locally decompose a given τ(t) ∈�k(N 5) for t ∈ (a, b) as
τ(t)= α(t)+β(t)∧ η(t),
where α(t) ∈∧k〈e1(t), . . . , e4(t)〉 and β(t) ∈
∧k−1〈e1(t), . . . , e4(t)〉. Therefore,
Jτ(t)= Jα(t)+ Jβ(t)∧ Jη(t)= Itα(t)+ Itβ(t)∧ dt
= Itτ(t)− (−1)k It(iξ(t)τ(t))∧ dt.
STRONG KÄHLER WITH TORSION STRUCTURES 67
Applying this to J dF , we get
J dF = J dω1− J (∂tω1+ dη)∧ η(t)
= It dω1− It(∂tω1+ dη)∧ η(t)+ It(iξ(t)dω1)∧ dt
− It(iξ (∂tω1+ dη))∧ η(t)∧ dt.
Finally, taking the differential of J dF , we get
d J dF = d(It dω1− It(∂tω1+ dη)∧ η(t)
)+ ∂t
(It dω1− It(∂tω1+ dη)∧ η(t)
)∧ dt
+ d(It(iξ(t)dω1)− It(iξ (∂tω1+ dη))∧ η(t)
)∧ dt. �
Remark 6.3. Observe that the first equation in (33) is exactly condition (28) forF = ω1(t)+ η(t)∧ dt . See Remark 5.2.
As a consequence of Lemma 6.1 and Proposition 6.2, we get the following:
Theorem 6.4. For t ∈ (a, b), let (η(t), ω1(t), ω2(t), ω3(t)) be a family of SU(2)-structures on a 5-dimensional manifold N 5 such that
(34) d(ω2(t)∧ η(t))= 0 and d(ω3(t)∧ η(t))= 0
for any t. If the evolution equations
(35)
d(It dω1(t)− It(∂tω1(t)+ dη(t))∧ η(t))= 0,
∂t(It dω1(t)− It(∂tω1(t)+ dη(t))∧ η(t))
=−d(It(iξ dω1(t))− It(iξ (∂tω1(t)+ dη(t)))∧ η(t)),
∂t(ω2(t)∧ η(t))=−dω3(t),
∂t(ω3(t)∧ η(t))= dω2(t),
are satisfied, then the SU(3)-structure on M = N × (a, b) given by
(36) F = ω1(t)+ η(t)∧ dt and 9 = (ω2(t)+ iω3(t))∧ (η(t)− idt)
is SKT.
Example 6.5. Consider the Lie algebra with structure equations
de j= 0 for = 1, 2, 3, de4
= e12, de5= e14,
which underlies the 5-dimensional nilmanifold N 5 considered in Section 5.1. Weendow it with the SU(2)-structure given by (30). It is easy to verify that
d(ω2 ∧ η)= d(ω3 ∧ η)= d(ω1 ∧ω1)= 0.
68 MARISA FERNÁNDEZ, ANNA FINO, LUIS UGARTE AND RAQUEL VILLACAMPA
We evolve this SU(2)-structure by
ω1(t)=−e14− e53, ω2(t)=−(1+ 3
2 t)1/3e15− (1+ 3
2 t)−1/3e34,
η(t)= (1+ 32 t)1/3e2, ω3(t)=−(1+ 3
2 t)1/3e13− (1+ 3
2 t)−1/3e45,
where t ∈ (−2/3,∞).For any t ∈ (−2/3,∞), the family (ω1(t), ω2(t), ω3(t), η(t)) satisfies equations
(34) and the last two equations of (35). Moreover, it satisfies the conditions
∂tω1(t)= 0, d(η(t))= 0, iξ (d(ω1(t)))= 0, ∂t(It(dω1(t)))= 0,
which implies that the evolution equations (33) are also satisfied.On the product N 5
×R, we consider the local basis of 1-forms
β1= (1+ 3
2 t)1/3e1, β2= (1+ 3
2 t)−1/3e4, β3= e5,
β4= e3, β5
= (1+ 32 t)1/3e2, β6
= dt.
The structure equations are
dβ1=−
12(1+
32 t)−1β16, dβ4
= 0,
dβ2= (1+ 3
2 t)−1(β15+
12β
26), dβ5=−
12(1+
32 t)−1β56,
dβ3= β12, dβ6
= 0.
Locally, J is given by Jβ1=−β2, Jβ3
=−β4 and Jβ5= β6. The fundamental
form F = −β12− β34
+ β56 satisfies d(J dF) = 0, and the (3, 0)-form 9 =
(β1+ iβ2)∧ (β3
+ iβ4)∧ (β5− iβ6) is closed. Therefore, (F, 9) is a local SKT
SU(3)-structure on N 5×R.
Remark 6.6. A Hermitian structure (J, h) on a 6-dimensional manifold M6 iscalled balanced if F∧F is closed, F being the associated fundamental 2-form. Thepaper [Fernandez et al. 2009] introduced the notion of balanced SU(2)-structureson 5-dimensional manifolds, together with appropriate evolution equations whosesolution gives rise to a balanced SU(3)-structure in six dimensions.
If M6 is compact, then a balanced structure cannot be SKT; see, for instance,[Fino et al. 2004].
The SU(2)-structure (30) from the previous example is also balanced, and itgives rise to a balanced metric on the product of N 5 with a open interval; see[Fernandez et al. 2009, (11)]. However, one can check directly that this solution isnot SKT.
If G is the nilpotent Lie group underlying N 5, the product G × R has no left-invariant SKT structures and does not admit any left-invariant complex structures;however, we can find a local SKT SU(3)-structure on it.
STRONG KÄHLER WITH TORSION STRUCTURES 69
7. HKT structures
We will now find conditions under which an S1-bundle over a (4n+3)-dimensionalmanifold endowed with three almost contact metric structures is hyper-Kahler withtorsion (HKT, for short). Recall that a 4n-dimensional hyper-Hermitian manifold(M4n, J1, J2, J3, h) is a hypercomplex manifold (M4n, J1, J2, J3) endowed with aRiemannian metric h compatible with the complex structures Jr for r = 1, 2, 3;that is, h satisfies
h(Jr X, Jr Y )= h(X, Y )
for any r = 1, 2, 3 and any vector fields X and Y on M4n .A hyper-Hermitian manifold (M4n, J1, J2, J3, h) is called HKT if and only if
(37) J1 dF1 = J2 dF2 = J3 dF3,
where Fr denotes the fundamental 2-form associated to the Hermitian structure(Jr , h); see [Grantcharov and Poon 2000].
We consider a (4n+3)-dimensional manifold N 4n+3 endowed with three almostcontact metric structures (Ir , ξr , ηr , g) for r = 1, 2, 3, and satisfying
(38)Ik = Ii I j − η j ⊗ ξi =−I j Ii + ηi ⊗ ξ j ,
ξk = Iiξ j =−I jξi , ηk = ηi I j =−η j Ii .
By applying Theorem 2.3, we can construct hyper-Hermitian structures on S1-bundles over N 4n+3 and study when they are strong HKT.
Theorem 7.1. Let N 4n+3 be a (4n+3)-dimensional manifold with three normalalmost contact metric structures (Ir , ξr , ηr , g) for r = 1, 2, 3, and satisfying (38).Let � be a closed 2-form on N 4n+3 that represents an integral cohomology class,and that is Ir -invariant for every r = 1, 2, 3. Consider the circle bundle S1 ↪→
P → N 4n+3 with a connection 1-form θ whose curvature form is dθ = π∗(�),where π : P→ N is the projection. The hyper-Hermitian structure (J1, J2, J3, h)on P defined by (2) and (4) is HKT if and only if
(39)
π∗(I1(dω1))−π∗(dη1)∧π
∗η1 = π∗(I2(dω2))−π
∗(dη2)∧π∗η2
= π∗(I3(dω3))−π∗(dη3)∧π
∗η3,
π∗(I1(iξ1dω1))= π∗(I2(iξ2dω2))= π
∗(I3(iξ3dω3)),
where ωr is the fundamental form of the almost contact structure (Ir , ξr , ηr , g).Moreover, the HKT structure is strong if and only if
(40)d(π∗(Ir (iξr dωr )))= 0,
d(π∗(Ir (dωr )− dηr ∧ ηr ))= (π∗(−Ir (iξr dωr ))+π
∗�)∧π∗�
for every r = 1, 2, 3.
70 MARISA FERNÁNDEZ, ANNA FINO, LUIS UGARTE AND RAQUEL VILLACAMPA
Proof. The almost hyper-Hermitian structure (J1, J2, J3, h) on P defined by (2)and (4) is hyper-Hermitian if and only (Ir , ξr , ηr , g) is normal and dθ is Jr -invariantfor every r = 1, 2, 3. The HKT condition is equivalent to (37). By (9), we have
Jr d Fr = π∗(Ir (dωr ))+π
∗(Ir (iξr dωr ))∧ θ −π∗(dηr )∧π
∗ηr − θ ∧ dθ,
where Fr is the fundamental 2-form of (Jr , h). Therefore, condition (37) is satisfiedif and only if (39) holds. Finally, the Jr dFr are closed if and only if (40) holds. �
On N 4n+3×R, consider the almost Hermitian structures (Jr , Fr , h) defined by
(41)h = g+ (dt)2, Fr = ωr + ηr ∧ dt,
Jr (ηr )= dt, Jr (X)= Ir (X) for X ∈ Ker ηr .
By (38), we have
J1 J2 = J3 =−J2 J1,
J1η2 = I1η2 =−η3, J2η3 = I2η3 =−η1, J3η1 = I3η1 =−η2.
Therefore, (Jr , Fr , h) for r = 1, 2, 3 is a hyper-Hermitian structure on N 4n+3×R
if and only if the structures (Ir , ξr , ηr , g) are normal.
Corollary 7.2. Let N 4n+3 be a (4n+3)-dimensional manifold endowed with threenormal almost contact metric structures (Ir , ξr , ηr , g) for r = 1, 2, 3. On theproduct N 4n+3
×R, consider the hyper-Hermitian structure (J1, J2, J3, h) definedby (41). Then, (J1, J2, J3, h) is HKT if and only if
I1(dω1)− dη1 ∧ η1 = I2(dω2)− dη2 ∧ η2 = I3(dω3)− dη3 ∧ η3,
I1(iξ1dω1)= I2(iξ2dω2)= I3(iξ3dω3).
The HKT structure is strong if and only if
d(Ir (iξr dωr ))= 0 and d(Ir (dωr )− dηr ∧ ηr )= 0 for every r = 1, 2, 3.
Moreover, if (J1, J2, J3, h) is such that dη1∧η1 = dη2∧η2 = dη3∧η3 and one ofthe conditions
(a) dωr = 0 for any r = 1, 2, 3, that is, (Ir , ξr , ηr , g) is quasi-Sasakian for anyr = 1, 2, 3; or
(b) dωi ∧ η j ∧ ηk 6= 0, where (i, j, k) is a permutation of (1, 2, 3), as well as
I1(dω1)= I2(dω2)= I3(dω3) and I1(iξ1dω1)= I2(iξ2dω2)= I3(iξ3dω3),
is satisfied, then (J1, J2, J3, h) is HKT. In case (a), the HKT structure is strong. Incase (b), the HKT structure is strong if and only if d(I1(dω1))= d(I1(iξ1dω1))= 0.
STRONG KÄHLER WITH TORSION STRUCTURES 71
Proof. By Theorem 7.1, the hyper-Hermitian structure (Jr , Fr , h) for r = 1, 2, 3 isHKT if and only if
(42)I1(dω1)− dη1 ∧ η1 = I2(dω2)− dη2 ∧ η2 = I3(dω3)− dη3 ∧ η3,
I1(iξ1dω1)= I2(iξ2dω2)= I3(iξ3dω3).
Locally, we write
(43) dωr = αr +
3∑i=1
βri ∧ ηi +
3∑i< j=1
γ ri j ∧ ηi ∧ η j + ρrη1 ∧ η2 ∧ η3,
where ρr are smooth functions, while αr , βri , and γ r
i j are respectively 3-forms,2-forms, and 1-forms in
⋂3i=1 Ker ηi .
By first using the normality of the three almost contact metric structures, andthen that iξr dηr = 0 and Ir (dηr )= dηr , locally we can write
(44)
dη1 = A1+ B1 ∧ η2− I1 B1 ∧ η3+C1η2 ∧ η3,
dη2 = A2+ B2 ∧ η1+ I2 B2 ∧ η3+C2η1 ∧ η3,
dη3 = A3+ B3 ∧ η1− I3 B3 ∧ η2+C3η1 ∧ η2,
where Ir Ar = Ar . Here, the Ar and Br are respectively 2-forms and 1-forms in⋂3i=1 Ker ηi , while the Cr are smooth functions. We have
Jr (dFr )= Jr (dωr )+ Jr (dηr ∧ dt)= Jr (dωr )− dηr ∧ ηr .
Therefore, by using (43) and (44), we obtain
J1(dF1)= I1α1+ I1β11 ∧ dt − A1 ∧ η1− I1β
13 ∧ η2− I1β
12 ∧ η3
− I1γ113 ∧ η2 ∧ dt + I1γ
112 ∧ η3 ∧ dt + B1 ∧ η1 ∧ η2− I1 B1 ∧ η1 ∧ η3
+ I1γ123 ∧ η2 ∧ η3+ ρ1η2 ∧ η3 ∧ dt −C1η1 ∧ η2 ∧ η3,
J2(dF2)= I2α2+ I2β22 ∧ dt − I2β
23 ∧ η1− A2 ∧ η2+ I2β
21 ∧ η3
+ I2γ223 ∧ η1 ∧ dt + I2γ
212 ∧ η3 ∧ dt − B2 ∧ η1 ∧ η2+ I2γ
213 ∧ η1 ∧ η3
+ I2 B2 ∧ η2 ∧ η3− ρ2η1 ∧ η3 ∧ dt +C2η1 ∧ η2 ∧ η3,
J3(dF3)= I3α3+ I3β33 ∧ dt + I3β
32 ∧ η1− I3β
31 ∧ η2− A3 ∧ η3
+ I3γ323 ∧ η1 ∧ dt − I3γ
313 ∧ η2 ∧ dt + I3γ
312 ∧ η1 ∧ η2− B3 ∧ η1 ∧ η3
+ I3 B3 ∧ η2 ∧ η3+ ρ3η1 ∧ η2 ∧ dt −C3η1 ∧ η2 ∧ η3.
72 MARISA FERNÁNDEZ, ANNA FINO, LUIS UGARTE AND RAQUEL VILLACAMPA
The conditions (42) are satisfied if and only if
(45)
γ 112 = γ
113 = γ
212 = γ
223 = γ
313 = γ
323 = 0,
ρr = 0, C1 =−C2 = C3,
I1α1 = I2α2 = I3α3, I1β11 = I2β
22 = I3β
33 ,
A1 = I2β23 =−I3β
32 , A2 =−I1β
13 = I3β
31 , A3 = I1β
12 =−I2β
21 ,
B1 =−B2 = I3γ312, −I1 B1 =−B3 = I2γ
213, I2 B2 = I3 B3 = I1γ
123.
Since Ir Ar = Ar the coefficients βri for r 6= i = 1, 2, 3 must satisfy the conditions
Ii (βij − Ikβ
ij )= 0 for all i, j, k = 1, 2, 3 with i 6= j , j 6= k and k 6= i.
The last three equations in (45) are satisfied if and only if γ 123 = γ
213 = γ
312 = 0.
Thus, finally, we obtain
(46)
dωr = αr +∑3
i=1 βri ∧ ηi , dηi = Ai + λ η j ∧ ηk,
0= Ii (βij − Ikβ
ij ) for all i, j, k = 1, 2, 3 with i 6= j , j 6= k and k 6= i,
I1α1 = I2α2 = I3α3,
A1 = I2β23 =−I3β
32 , A2 =−I1β
13 = I3β
31 , A3 = I1β
12 =−I2β
21
for any even permutation of (1, 2, 3).The expression for d(J1dF1) is
d(J1d F1)= d(I1(dω1)+ I1(iξ 1dω1)∧ dt)− d((dη1)∧ η1)
= d(I1(dω1))+ d(I1(iξ 1dω1))∧ dt − dη1 ∧ dη1
= d(I1(dω1)− dη1 ∧ η1)+ d(I1(iξ 1dω1))∧ dt,
and thus the HKT structure is strong if and only if
d(I1(dω1)− dη1 ∧ η1)= 0 and d(I1(iξ 1dω1))= 0.
To prove the last part of the corollary it suffices to consider coefficients β ir = 0 if
r 6= i in expression (43). �
Example 7.3. Consider the 7-dimensional Lie group G = SU(2)nR4, with struc-ture equations
de1=−
12 e25−
12 e36−
12 e47, de5
= e67,
de2=
12 e15+
12 e37−
12 e46, de6
=−e57,
de3=
12 e16−
12 e27+
12 e45, de7
= e56.
de4=
12 e17+
12 e26−
12 e35,
STRONG KÄHLER WITH TORSION STRUCTURES 73
By [Fino and Tomassini 2008], G admits a compact quotient M7=0\G by a uni-
form discrete subgroup 0, and is endowed with a weakly generalized G2-structure.By [Barberis and Fino 2008], M7
× S1 admits a strong HKT structure. We canshow that M7 has three normal almost contact metric structures (Ir , ξr , ηr , g) forr = 1, 2, 3 that are given by
I1e1= e2, I1e3
= e4, I1e5= e6, η1 = e7,
I2e1= e3, I2e2
=−e4, I2e5=−e7, η2 = e6,
I3e1= e4, I3e2
= e3, I3e6= e7, η3 = e5
and that satisfy the conditions of Corollary 7.2(a).
References
[Andrada et al. 2009] A. Andrada, A. Fino, and L. Vezzoni, “A class of Sasakian 5-manifolds”,Transformation Groups 14:3 (2009), 493–512. MR 2534796 Zbl 1185.53046 arXiv 0807.1800
[Barberis and Fino 2008] M. L. Barberis and A. Fino, “New HKT manifolds arising from quater-nionic representations”, preprint, 2008. To appear in Math. Z. arXiv 0805.2335
[Barberis et al. 2009] M. L. Barberis, I. G. Dotti, and M. Verbitsky, “Canonical bundles of complexnilmanifolds, with applications to hypercomplex geometry”, Math. Res. Lett. 16:2 (2009), 331–347.MR 2010g:53083 Zbl 1178.32014
[Bismut 1989] J.-M. Bismut, “A local index theorem for non-Kähler manifolds”, Math. Ann. 284:4(1989), 681–699. MR 91i:58140
[Blair 1967] D. E. Blair, “The theory of quasi-Sasakian structures”, J. Differential Geometry 1(1967), 331–345. MR 37 #2127 Zbl 0163.43903
[Blair 2002] D. E. Blair, Riemannian geometry of contact and symplectic manifolds, Progress inMathematics 203, Birkhäuser, Boston, MA, 2002. MR 2002m:53120 Zbl 1011.53001
[Boyer and Galicki 1999] C. Boyer and K. Galicki, “3-Sasakian manifolds”, pp. 123–184 in Surveysin differential geometry: Essays on Einstein manifolds, edited by C. LeBrun and M. Wang, Surv.Differ. Geom. 6, Int. Press, Boston, 1999. MR 2001m:53076 Zbl 1008.53047
[Conti and Salamon 2007] D. Conti and S. Salamon, “Generalized Killing spinors in dimension 5”,Trans. Amer. Math. Soc. 359:11 (2007), 5319–5343. MR 2008h:53077 Zbl 1130.53033
[Egidi 2001] N. Egidi, “Special metrics on compact complex manifolds”, Differential Geom. Appl.14:3 (2001), 217–234. MR 2002b:32038 Zbl 0981.32014
[Fernández et al. 2009] M. Fernández, A. Tomassini, L. Ugarte, and R. Villacampa, “Balanced Her-mitian metrics from SU(2)-structures”, J. Math. Phys. 50:3 (2009), 033507, 15. MR 2010f:53124Zbl 05646541
[Fino and Grantcharov 2004] A. Fino and G. Grantcharov, “Properties of manifolds with skew-symmetric torsion and special holonomy”, Adv. Math. 189:2 (2004), 439–450. MR 2005k:53062Zbl 1114.53043
[Fino and Tomassini 2008] A. Fino and A. Tomassini, “Generalized G2-manifolds and SU(3)-structures”, Internat. J. Math. 19:10 (2008), 1147–1165. MR 2009m:53054 Zbl 1168.53012
[Fino and Tomassini 2009] A. Fino and A. Tomassini, “Blow-ups and resolutions of strong Kählerwith torsion metrics”, Adv. Math. 221:3 (2009), 914–935. MR 2010d:32012
74 MARISA FERNÁNDEZ, ANNA FINO, LUIS UGARTE AND RAQUEL VILLACAMPA
[Fino et al. 2004] A. Fino, M. Parton, and S. Salamon, “Families of strong KT structures in sixdimensions”, Comment. Math. Helv. 79:2 (2004), 317–340. MR 2005f:53061 Zbl 1062.53062
[Friedrich and Ivanov 2002] T. Friedrich and S. Ivanov, “Parallel spinors and connections withskew-symmetric torsion in string theory”, Asian J. Math. 6:2 (2002), 303–335. MR 2003m:53070Zbl 1127.53304
[Gates et al. 1984] S. J. Gates, Jr., C. M. Hull, and M. Rocek, “Twisted multiplets and new super-symmetric nonlinear σ -models”, Nuclear Phys. B 248:1 (1984), 157–186. MR 87b:81108
[Gauduchon 1984] P. Gauduchon, “La 1-forme de torsion d’une variété hermitienne compacte”,Math. Ann. 267:4 (1984), 495–518. MR 87a:53101 Zbl 0523.53059
[Grantcharov and Poon 2000] G. Grantcharov and Y. S. Poon, “Geometry of hyper-Kähler connec-tions with torsion”, Comm. Math. Phys. 213:1 (2000), 19–37. MR 2002a:53059
[Grantcharov et al. 2008] D. Grantcharov, G. Grantcharov, and Y. S. Poon, “Calabi–Yau connec-tions with torsion on toric bundles”, J. Differential Geom. 78:1 (2008), 13–32. MR 2009g:32052Zbl 1171.53044
[Hitchin 2001] N. Hitchin, “Stable forms and special metrics”, pp. 70–89 in Global differentialgeometry: The mathematical legacy of Alfred Gray (Bilbao, 2000), edited by M. Fernández andJ. A. Wolf, Contemp. Math. 288, Amer. Math. Soc., Providence, RI, 2001. MR 2003f:53065Zbl 1004.53034
[Howe and Papadopoulos 1996] P. S. Howe and G. Papadopoulos, “Twistor spaces for hyper-Kählermanifolds with torsion”, Phys. Lett. B 379:1-4 (1996), 80–86. MR 97h:53073
[Ivanov and Papadopoulos 2001] S. Ivanov and G. Papadopoulos, “Vanishing theorems and stringbackgrounds”, Classical Quantum Gravity 18:6 (2001), 1089–1110. MR 2002h:53076 Zbl 0990.53078
[Kobayashi 1956] S. Kobayashi, “Principal fibre bundles with the 1-dimensional toroidal group”,Tôhoku Math. J. (2) 8 (1956), 29–45. MR 18,328a Zbl 0075.32103
[Ogawa 1963] Y. Ogawa, “Some properties on manifolds with almost contact structures”, TôhokuMath. J. (2) 15 (1963), 148–161. MR 27 #704 Zbl 0147.41002
[Sasaki and Hatakeyama 1961] S. Sasaki and Y. Hatakeyama, “On differentiable manifolds withcertain structures which are closely related to almost contact structure, II”, Tôhoku Math. J. (2) 13(1961), 281–294. MR 25 #1513 Zbl 0112.14002
[Strominger 1986] A. Strominger, “Superstrings with torsion”, Nuclear Phys. B 274:2 (1986), 253–284. MR 87m:81177
[Swann 2010] A. Swann, “Twisting Hermitian and hypercomplex geometries”, Duke Math. J. 155(2010), 403–431.
[Ugarte 2007] L. Ugarte, “Hermitian structures on six-dimensional nilmanifolds”, TransformationGroups 12:1 (2007), 175–202. MR 2008e:53139 Zbl 1129.53052
Received September 23, 2009.
MARISA FERNÁNDEZ
UNIVERSIDAD DEL PAÍS VASCO
DEPARTAMENTO DE MATEMÁTICAS
FACULTAD DE CIENCIA Y TECNOLOGÍA
APARTADO 64448080 BILBAO
SPAIN
STRONG KÄHLER WITH TORSION STRUCTURES 75
ANNA FINO
UNIVERSITÀ DI TORINO
DIPARTIMENTO DI MATEMATICA
VIA CARLO ALBERTO 10I-10123 TORINO
ITALY
LUIS UGARTE
UNIVERSIDAD DE ZARAGOZA
DEPARTAMENTO DE MATEMÁTICAS-IUMACAMPUS PLAZA SAN FRANCISCO
50009 ZARAGOZA
SPAIN
RAQUEL VILLACAMPA
CENTRO UNIVERSITARIO DE LA DEFENSA
ACADEMIA GENERAL MILITAR
CRTA. DE HUESCA S/N
50090 ZARAGOZA
SPAIN
PACIFIC JOURNAL OF MATHEMATICShttp://www.pjmath.org
Founded in 1951 byE. F. Beckenbach (1906–1982) and F. Wolf (1904–1989)
EDITORS
Vyjayanthi ChariDepartment of Mathematics
University of CaliforniaRiverside, CA 92521-0135
Robert FinnDepartment of Mathematics
Stanford UniversityStanford, CA [email protected]
Kefeng LiuDepartment of Mathematics
University of CaliforniaLos Angeles, CA 90095-1555
V. S. Varadarajan (Managing Editor)Department of Mathematics
University of CaliforniaLos Angeles, CA 90095-1555
Darren LongDepartment of Mathematics
University of CaliforniaSanta Barbara, CA 93106-3080
Jiang-Hua LuDepartment of Mathematics
The University of Hong KongPokfulam Rd., Hong Kong
Alexander MerkurjevDepartment of Mathematics
University of CaliforniaLos Angeles, CA 90095-1555
Sorin PopaDepartment of Mathematics
University of CaliforniaLos Angeles, CA 90095-1555
Jie QingDepartment of Mathematics
University of CaliforniaSanta Cruz, CA 95064
Jonathan RogawskiDepartment of Mathematics
University of CaliforniaLos Angeles, CA 90095-1555
Silvio Levy, Scientific Editor Matthew Cargo, Senior Production Editor
SUPPORTING INSTITUTIONS
ACADEMIA SINICA, TAIPEI
CALIFORNIA INST. OF TECHNOLOGY
INST. DE MATEMÁTICA PURA E APLICADA
KEIO UNIVERSITY
MATH. SCIENCES RESEARCH INSTITUTE
NEW MEXICO STATE UNIV.OREGON STATE UNIV.
STANFORD UNIVERSITY
UNIV. OF BRITISH COLUMBIA
UNIV. OF CALIFORNIA, BERKELEY
UNIV. OF CALIFORNIA, DAVIS
UNIV. OF CALIFORNIA, LOS ANGELES
UNIV. OF CALIFORNIA, RIVERSIDE
UNIV. OF CALIFORNIA, SAN DIEGO
UNIV. OF CALIF., SANTA BARBARA
UNIV. OF CALIF., SANTA CRUZ
UNIV. OF MONTANA
UNIV. OF OREGON
UNIV. OF SOUTHERN CALIFORNIA
UNIV. OF UTAH
UNIV. OF WASHINGTON
WASHINGTON STATE UNIVERSITY
These supporting institutions contribute to the cost of publication of this Journal, but they are not owners or publishers and have noresponsibility for its contents or policies.
See inside back cover or www.pjmath.org for submission instructions.
The subscription price for 2011 is US $420/year for the electronic version, and $485/year for print and electronic.Subscriptions, requests for back issues from the last three years and changes of subscribers address should be sent to Pacific Journal ofMathematics, P.O. Box 4163, Berkeley, CA 94704-0163, U.S.A. Prior back issues are obtainable from Periodicals Service Company,11 Main Street, Germantown, NY 12526-5635. The Pacific Journal of Mathematics is indexed by Mathematical Reviews, ZentralblattMATH, PASCAL CNRS Index, Referativnyi Zhurnal, Current Mathematical Publications and the Science Citation Index.
The Pacific Journal of Mathematics (ISSN 0030-8730) at the University of California, c/o Department of Mathematics, 969 EvansHall, Berkeley, CA 94720-3840, is published monthly except July and August. Periodical rate postage paid at Berkeley, CA 94704,and additional mailing offices. POSTMASTER: send address changes to Pacific Journal of Mathematics, P.O. Box 4163, Berkeley, CA94704-0163.
PJM peer review and production are managed by EditFLOW™ from Mathematical Sciences Publishers.
PUBLISHED BY PACIFIC JOURNAL OF MATHEMATICSat the University of California, Berkeley 94720-3840
A NON-PROFIT CORPORATIONTypeset in LATEX
Copyright ©2011 by Pacific Journal of Mathematics
PACIFIC JOURNAL OF MATHEMATICS
Volume 249 No. 1 January 2011
1Metabelian SL(n, C) representations of knot groups, II: Fixed pointsHANS U. BODEN and STEFAN FRIEDL
11Lewis–Zagier correspondence for higher-order formsANTON DEITMAR
23Topology of positively curved 8-dimensional manifolds with symmetryANAND DESSAI
49Strong Kähler with torsion structures from almost contact manifoldsMARISA FERNÁNDEZ, ANNA FINO, LUIS UGARTE and RAQUEL
VILLACAMPA
77Connections between Floer-type invariants and Morse-type invariants of Legendrianknots
MICHAEL B. HENRY
135A functional calculus for unbounded generalized scalar operators on Banach spacesDRAGOLJUB KECKIC and ÐORÐE KRTINIC
157Geometric formality of homogeneous spaces and of biquotientsD. KOTSCHICK and S. TERZIC
177Positive solutions for a nonlinear third order multipoint boundary value problemYANG LIU, ZHANG WEIGUO, LIU XIPING, SHEN CHUNFANG and CHEN
HUA
189The braid group surjects onto G2 tensor spaceSCOTT MORRISON
199Analogues of the Wiener Tauberian and Schwartz theorems for radial functions onsymmetric spaces
E. K. NARAYANAN and ALLADI SITARAM
211Semidirect products of representations up to homotopyYUNHE SHENG and CHENCHANG ZHU
237Homology sequence and excision theorem for Euler class groupYONG YANG
255Acknowledgement
0030-8730(201101)249:1;1-D
PacificJournalofM
athematics
2011Vol.249,N
o.1